Name HW 6 - Center of Mass Homework Date Imagine that you have a rectangular prism with a square cross section (length L, width and height are each equal to W). The density of this object varies linearly along the length. (It was manufactured by placing small lead beads into a plastic matrix and the beads settled as the matrix hardened.) The density at the right side is Po and increases towards the left to a nal value of 290. Use a coordinate system with +x to the right, +y upwards, and +2 outwards. 290 pm a: l' U 4! L A) Write an expression for the density, p, as a function of horizontal position x. Take x = 0 to be the left side of the prism. B) What is the volume dV ofa slice oflength dx? C) Imagine that you have a small chunk of the material located a distance x from the right end and with a length of dx (and width and height of W). Write a symbolic expression for the mass of this chunk (dm) as a function of p(x) and other quantities. D) Determine the total mass of the object in two ways (reasoning and integral). Show your work and justify your answers. E) How many dimensions is this integral (1, 2 or 3)? Explain why. F) In terms of the y (up towards the top of the page) and 2 (out of the page sort of), where is the center of mass? Justify your answer. G) Recall that the basic form for the location of the center of mass is Pam = I T dm . Based on your fdm previous answers, nd the location of the center of mass